Milan mathematical biology papers
Matylda Jabłońska
Lappeenranta University of Technology
Lappeenranta, 28.01.2009
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Agenda
1
Paper 1: Tumour-induced angiogenesis
Modelling problem
Modelling approach
2
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Vincenco Capasso, Daniela Morale
"Stochastic modelling of tumour-induced angiogenesis."
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
"The mathematical modelling of an angiogenic process derives from
the strong coupling of the kinetic parameters of the relevant
stochastic branching-and-growth of the capillary network with a
family of interacting underlying fields. The aim of this paper is to
propose a novel mathematical approach for reducing complexity by
(locally) averaging the stochastic cell, or vessel densities in the
evolution equations of the underlying fields, at the mesoscale, while
keeping stochasticity at lower scales, possibly at the level of
individual cells or vessels. (. . . ) The branching mechanism of blood
vessels is modelled as a stochastic marked counting process
describing the branching of new tips, while the network of vessels is
modelled as the union of the trajectories developed by tips, according
to a system of stochastic differential equations á la Langevin."
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
"The normal, healthy body maintains a perfect balance of
angiogenesis modulators. In many serious diseases states, the body
looses control over angiogenesis. Angiogenesis-dependent diseases
result when new blood vessels either grow excessively or
insufficiently. Nowadays angiogenesis is very widespread studied in
relation with the growth of tumours."
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Network dynamics:
vessel branching;
vessel extension;
chemotaxis in response to a generic tumour angiogenetic factor
(TAF), released by tumour cells;
haptotatic migration in response to fibronectin gradient,
emerging from the extracellular matrix and through degradation
and production by endothelial cells themselves;
anastomosis, when a capillary tip meets an existing vessel.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Let N0 denote the initial number of tips, N(t) the number of tips at
time t, and X i (t) the location of the i-th tip at time t. We model
sprout extension by tracking the trajectory of each individual
capillary tips. As a consequence
N(t)
X (t) =
[
{X i (s), Ti ≤ s ≤ t}
i=1
will be the network of endothelial cells, i.e. the union of trajectories
of the tips, where Ti is the birth time of the i-th tip and
N(t)
Y (t) =
[
{X i (s), t̃1 ≤ t − s ≤ t̃2 }
i=1
is the union of the mature parts of vessels that may branch at time t.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Given a TAF’s concentration C (t, x)
tip branching occurs with rate per unit of volume:
α1 (t, x) = α1 β1 (C (t, x))
N(t)
X
δX i (t) (x)
i=1
vessel branching occurs with rate per unit of volume:
α2 (t, x) = α2 β2 (C (t, x))δY (t) (x)
probability of having a new tip:


Z
N(t)
X
P =
α1 (t, X i (t)) + α2 (x, t)dx  dt =: αN (t)dt
i=1
Rd
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
As far as vessel movement (extension) is concerned, we consider a
Langevin model:
dX i (t) = v i (t)(1 − pa IX (t) (X k (t)))dt, t > T i ,
dv i (t) = a(X i (t), v i (t), t)dt + σdW i (t), t > T i ,
where v i (t) is the velocity of the i-th tip at time t. The drift
function a(x, v , t) is a function of the concentration C (t, x) of TAF.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Chemotatic field:
∂
C (t, x) = c1 δA (x) + d1 ∆C (t, x)
∂t
N(t)
1 X i
(v (t)δX i (t) ∗ V )(x)
− ηC (t, x)
N i=1
Haptotatic field:
N(t)
∂
1 X
f (t, x) = β
(δ i ∗ V )(x) − γm(t, x)f (t, x).
∂t
N i=1 X (t)
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
The chemotatic and haptotatic fields depend upon the stochastic
geometric process X (t) of the vessel network. A direct consequence
is the stochasticity of the underlying fields, and consequently the
stochasticity of the kinetic parameters of birth and growth of vessels.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Figure: A vessel network (on the left) driven by a constant in time TAF
field (on the right).
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approach
Figure: A vessel network (on the left) interacting with a degrading TAF
field (on the right).
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Daniela Morale, Vincenco Capasso, Karl Oelschlager
"An interacting particle system modelling aggregation behavior:
from individuals to populations."
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Modelling problem
"Complexity of the biological clustering raises very interesting
mathematical problems. Aggregation patterns are usually explained
in terms of forces, external and/or internal, acting upon individuals.
A remarkable aspect of these global organization is that individuals
move altogether in a coordinated (though random) fashion. (...)
The aim of the modelling is to catch the main features of the
interaction at the lower scale of single individuals that are
responsible, at a larger scale, for a more complex behavior that leads
to the formation of aggregating patterns."
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Eulerian approach
Eulerian models are based on continuum equations, typically
(deterministic) nonlinear partial differential equations of the
advection-reaction-diffusion type:
ρt + ∇ · (vρ) = ∇ · (D∇ρ) + ν(ρ)
where ρ is the population density, v is the velocity field and ν(ρ) is a
possible additive reaction term which may include birth and death
processes.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Lagrangian approach
In Lagrangian approach individuals are followed in their motion.
Variation in time of the (random) location of the k-th individual in
the group at time t ≥ 0, XNk (t) ∈ Rd , k = 1, . . . , N is described by
a system of stochastic differential equations:
dXNk (t) = FN [XN (t)] XNk (t) dt + σN dW k (t), k = 1, . . . , N,
where each particle moves randomly with a mean free path σN . FN
is drift term with two additive components responsible for
aggregation and repulsion.
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
The interaction among particles is mathematically modelled by an
interaction kernel depending on the distance between two particles.
The main three types of interactions are:
McKean-Vlasov interaction (macroscale)
hydrodynamic interaction (microscale)
moderate interaction (mesoscale)
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
The three types of interaction may be obtained in terms of an
appropriate rescaling of a given reference function V1 . Let V1 be a
sufficiently regular probability density; and assume that the
interaction of two particles out of N, located in x and y respectively
is modelled by
1
VN (x − y )
N
where
VN (z) = N β V1 (N β/d z),
which expresses the rescaling of V1 with respect to the total number
N of particles in terms of a scaling coefficient β ∈ [0, 1].
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
The force exerted on the k-th (out of N) single particle located at
XNk (t) due to the interaction of the single k-particle with all the
others in the population is given by
k
I ≡I
k
(XN1 (t), . . . , XNN (t))
N
X
1
=
VN XNk (t) − XNi (t)
N
i=1
=
N
X
N β−1 V1 N β/d XNk (t) − XNi (t)
i=1
where we obtain a McKean-Vlasov interaction for β = 0, a
hydrodynamic interaction for β = 1, and a moderate interaction for
β ∈ (0, 1).
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
Comparison of approaches
Eulerian models:
Identity of individuals is compromised.
Some important features of the dynamics may be hidden.
Lagrangian models:
Stochastic Lagrangian models offer the advantage of being
directly related to experimental data on the behavior of
individuals of a real population, especially when dealing with a
relatively "small" number of individuals per unit space (Poisson
like spatial processes).
Matylda Jabłońska
Milan mathematical biology papers
Paper 1: Tumour-induced angiogenesis
Paper 2: Modelling population aggregation behavior
Modelling problem
Modelling approaches
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Matylda Jabłońska
Milan mathematical biology papers